Mathematical cube puzzle

ABSTRACT

A puzzle made up of four separate cubes which are arrangeable into a plurality of differing solutions is described. Each edge of each cube has one of a set of three colors associated therewith, and the resulting arrangement or combination of colors around the edges on each cube face is different from that on every other cube face, and the colors of all common edges of adjacent faces on each individual cube match one another. The object of the puzzle is to arrange the various cubes relative to one another so that the colors associated with all exposed adjacent playing edges of different cubes match one another, e.g., are the same. Various solutions of differing degrees of difficulty are described satisfying this criteria.

United States Patent [191 Nelson 0 MATHEMATICAL CUBE PUZZLE [21] Appl.N0.: 258,872

[52] US. Cl. 273/156, 273/137 D [51] Int. Cl. A63f 9/08 [58] Field ofSearch 273/157 R, 156, 137 D [56] References Cited UNITED STATES PATENTS12/1970 Williams et al. 273/157 R 2/1865 Harold 273/157 R UX OTHERPUBLICATIONS New Mathematical Pastimes by P. A. MacMahon, published 1921by Cambridge University press.

[ Jan. 29, 1974 Primary Examiner-Anton O. Oechsle Attorney, Agent, orFirmMoore, Zimmerman &

Dubb

[ ABSIRACT A puzzle made up of four separate cubes which are arrangeableinto a plurality of differing solutions is described. Each edge of eachcube has one of a set of three colors associated therewith, and theresulting arrangement or combination of colors around the edges on eachcube face is different from that on every other cube face, and thecolors of all common edges of adjacent faces on each individual cubematch one another. The object of the puzzle is to arrange the variouscubes relative to one another so that the colors associated with allexposed adjacent playing edges of different cubes match one another,e.g., are the same. Various solutions of differing degrees of difficultyare described satisfying this criteria.

5 Claims, 3 Drawing Figures MATHEMATICAL CUBE PUZZLE BACKGROUND OF THEINVENTION The present invention relates to a mathematical puzzle and,more particularly, to such a puzzle made up of a number of similarthree-dimensional playing pieces, each of which has a plurality of facesdefining playing edges with which different indicia are associated. Theinvention further relates to an arrangement of such pieces into ageometrical pattern in which the indicia associated with adjacentplaying edges on different playing pieces match one another in apredetermined manner.

As the role of science and technology in our society has grown, gamesand puzzles which test the skill of a player in mathematics or logichave become increasingly popular. For example, a puzzle of this naturemarketed under the trademark Instant Insanity was quite widely acceptedby the general public at the time of its introduction to the market.This puzzle comprises four separate cubes having faces of differentcolors. The object of the puzzle is to align all four cubes in a rowwith the cubes so oriented relative to one another that the cube facesdefining each side of the resulting rectangular structure have apredetermined regular relationship. The arrangement on each cube of thedifferently colored faces is such, relative to the arrangement on thefaces on the other cubes, that only one combination of specificorientations of the cubes relative to one another provides the desiredsolution. It will be appreciated that because each cube had sixdifferent faces and could itself be arranged in numerous orientations inspace, the number of combinations of various possible cube orientationsis exceedingly high. The result is that the possibility of one findingthe solution via a trial and error method is quite low. However, as apractical matter, all attempts to find the solution are limited to doingso by trial and error, unless the potential solver is mathematicallytrained and has had experience with mathematical games so that he candiscover the mathematical relationships of the cube faces to one anotherand use this information in arriving at the solution.

After the initial popularity of the Instant Insanity puzzle at the timeit was introduced on the market, its popularity waned markedly. It isbelieved that one of the major reasons for this decline in market appealis that the puzzle has only one mode of solution. That is, after aplayer has found the on solution, the puzzle is not, in general, anylonger of interest to him. Moreover, as mentioned before, the finding ofthe solution is exceedingly difficult for the average player. Thus, manyplayers have become frustrated and lost interest in the puzzle beforediscovering the solution.

SUMMARY OF THE INVENTION The present invention relates to athree-dimensional puzzle of the Instant Insanity type which has aplurality of different solutions, ranging from solutions which arerelatively easy to find to those which are exceedingly difficult tofind. Thus, the puzzle is challenging to potential players of varyingskills. Because the puzzle has more than one mode of solution, thefinding by the player of any one solution will not automatically takeaway the stimulation provided by the puzzle. Furthermore, for some ofthe desired configurations a logical approach is available, as distinctfrom the trial and error method, which when recognized can lead rapidlyto the solution.

In its basic aspects, the puzzle of the invention includes a pluralityof similar three-dimensional pieces, e.g., cubes, having faces definingplaying edges with which indicia, such as colors, are associated. Thenumber of difierent indicia on each face of each piece varies, dependingupon the number of palying edges defined by the face and the particulararrangement or combination of the indicia represented by the edges onsuch face. This association of the indicia with the edges of the piece,rather than merely with the faces themselves as in the past, lendssubstantial versatility to the arrangement and makes a plurality ofpuzzle solutions, all satisfying the same puzzle criteria, possible.Most desirably, the order of the indicia associated with the edges oneach face represents a different one of the mathematical combinationsinto which the indicia can be placed in a circular or closed orderaround the edges. The result is that each of the faces differs fromevery other face in the puzzle so that there is generally only one wayin which any particular solution of the puzzle can be achieved.

The invention further includes an arrangement of such pieces into ageometrical pattern which satisfies certain criteria providing thedesired plurality of puzzle solutions. Broadly, such arrangement is onein which the indicia which is associated with adjacent ones of theplaying edges on different ones of the playing pieces match one anotherin a predetermined manner. Because the edges on each face thus play apart in defining a solution to the puzzle, solutions are made availablewhich are not straight line solutions, i.e., solutions are possible inwhich the playing pieces are not necessarily aligned in a row one afteranother. Rather, solutions to the puzzle include various arrangements ofthe pieces into different geometrical patterns.

The invention will be better understood and additional features andadvantages thereof will become apparent from the following more detaileddescription of a preferred embodiment.

BRIEF DESCRIPTION OF THE DRAWING With reference to the accompanyingsingle sheet of drawing:

FIG. 1 is an isometric view illustrating four cubes providing apreferred embodiment of the invention, three of such cubes being shownin the proper relationship to one another to provide one solution of theinvention, and the fourth cube in the process of being positioned tocomplete the solution;

FIG. 2 illustrates two-dimensional representations of each of the fourcubes of the preferred embodiment of FIG. 1; and

FIG. 3 illustrates twelve different arrangements of the cubes of FIG. 1which also satisfy the criteria for solutions to the puzzle of theinvention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT With reference to theaccompanying drawing, a preferred embodiment of the puzzle of theinvention is generally referred to by the reference numeral 11. As isillustrated, puzzle 11 is comprised of a plurality of similarthree-dimensional playing pieces. There are four of such pieces, andthey are in the form of cubes l2, l3, l4 and 16. Each of the cubes 12 16has a plurality of square faces providing playing edges for defining thepuzzle of the invention and the criteria by which the solutions to thepuzzle are governed. More particularly, with reference to cube 12, forexample, it is provided with six different square faces, one of which isreferred to by the reference numeral 17. Such face 17 has four edges,18, 19, 21 and 22, respectively. Each of said edges 18 22 is actuallydefined by the intersection of the face 17 with its adjacent facesmaking up the cube 12, with the result that each of the edges 18 22 isin fact an edge which is common to the face 17 and a respective one ofthe other faces of such cube.

As a particularly salient feature of the instant invention, each playingedge provided by a cube face has one of a set of indicia associatedtherewith. The indicia in this embodiment are different colors which areapplied to the respective faces so as to be associated with the playingedges thereof. More particularly, each face on a cube is divided intodifferently colored quadrants, each of which quadrants is subtended bythe playing edge with which the color is associated. In this particularinstance, each of the colors is selected from the set of three colorsyellow, red and green. For simplicity, such colors are represented inthe drawing by their respective initials.

The order of the colors around the edges of each of the faces representsone of the mathematical combinations into which such colors arearrangeable in a circular or closed order. By circular or closed orderis meant an order which has neither a definite beginning nor end. Sincethere are three colors in the set of colors, and four edges on eachface, one of such colors will appear at least twice on each face.

The combination into which the colors are arranged in a circular orderon each face is different for every face. That is, the order of thecolors on each of the faces represents a different one of the possiblecombinations into which the three colors can be arranged in a circularorder when taken four at a time. There are 24 different combinations inwhich such three colors can be so arranged in a circular order, whentaken four at a time. Moreover, since each cube has six faces and thereare four of such cubes in the puzzle, there are a total of 24 faces inthe puzzle. The result is that the plurality of faces provided by thecubes of the preferred embodiment represents a complete complement orset of the faces. In set theory terms, this plurality of facesrepresents one realization of the set of all possible way of choosing,with replacement, from a set of three distinct objects, ordered sets offour objects which remain distinct under any cyclic permutation.

Most desirably, the faces are so arranged on the cubes relative to oneanother that the color associated with each playing edge is the same onboth edges of each cube face where they intersect and have such edge incommon. For example, with respect to the cube 16 shown in FIG. 1, boththe upper face 23 and righthand side face 24 thereof have the coloryellow in the quadrants subtended by their common playing edge 26. Thisfeature of the puzzle reduces the possibility of confusion as to whichface controls an edge, i.e., which face determines the color to beassociated with the common edge.

There are various different arrangements of the cube faces relative toone another, which will result in the colors on adjacent facesassociated with the common edge therebetween being the same. However, apreferred arrangement satisfying this criteria is illustrated in FIG. 2which is a two-dimensional representation of the four cubes of FIG. 1.That is, the four figures of FIG. 2 show the six faces of the cubes theyrespectively represent, in the relationship such faces would have if thecubes were unfolded along edges as shown and then flattened. The samereference numerals are used in both FIGS. 1 and 2 to refer to likeparts, except such reference numerals are primed in FIG. 2.

The arrangement of cube faces depicted in FIG. 2,,is an arrangementdiscovered by the inventor which results in the puzzle having a solutionfor the geometrical pattern shown in FIG. I, in which the four cubes arealigned, one after another in a row. In this face arrangement, oppositefaces on two of the cubes are mirror images of corresponding faces onother cubes. One of such opposite faces on one of the two cubes is themirror image of one of the opposite faces on the other of such twocubes. More particularly, with reference to FIG. 2, face 27 on cuberepresentation 12 is the mirror image of face 28 of cube representation13'. Moreover, the face on cube l3 opposite to face 28', i.e., face 29',is the mirror image of the face 31 of cube representation 14'. The faceof cube l4 opposite face 31', i.e., face 32 is, in turn, the mirrorimage of face 33 of cube representation 16. Thus, each of the two cubesl3 and 14 have opposite faces, represented respectively at 28' and 29',and 31' and 32 in FIG. 2, which are mirror images of faces on othercubes. One of those faces which is a mirror image on cube 13, the facerepresented at 29, is a mirror image of one of the mirror image faces ofcube 14, i.e., the face represented at 31. The other mirror image facesrepresented at 28 and 32' on each of the cubes l3 and 14 are,respectively, mirror images of the faces of cubes l2 and 16 representedby faces 27 and 33 in FIG. 2. This mirror image face relationship of thecubes and the criteria that the colors subtended by a common edge be thesame on both faces defining such edge are the crucial concepts indetermining the relative placement of each of the faces on the cubes soas to provide a solution for the FIG. I configuration.

As another important aspect of the instant invention, it includes thecriteria defining the solution to the puzzle. More particularly, itincludes the arrangement of the cubes into a geometrical pattern inwhich the colors associated with adjacent edges on the adjacent cubesare the same. Most desirably, the pattern is one in which the cubes joinone another, either by edges touching or by opposed faces of adjacentcubes abutting one another. In such an arrangement, it is desirable thatall adjacent playing edges of those faces on different cubes which areexposed, i.e., not hidden in the pattern, be the same color. In thisconnection, the criteria determining a solution to the puzzle ispreferably considered met only when the criteria is satisfied by allexposed faces and edges, irrespective of the angle from which thepattern is viewed. In other words, the criteria should be satisfied evenfor the bottom surfaces of the pattern. However, if the faces definingan edge are all hidden within the pattern by being abutted against facesof other cubes, it is preferable that such edge need not satisfy anyparticular criteria. This will assure that the puzzle can have aplurality of solutions.

As mentioned previously, the solution depicted in FIG. I is possibleonly when the various cube faces are arranged in the relationshiprelative to one another described above. With reference to FIG. 1, itwill be seen that all visible, adjacent edges of different cubes in therow have the same color associated therewith. The faces shown in FIG. 1also uniquely define the orientations of all the cubes, and the adjacentedges of all exposed faces which are not depicted in FIG. 1 will alsohave the same colors associated therewith. That is, the edges on therear side and the bottom of the arrangement also meet the adjacent edgecriteria.

It will be further noted on considering the solution depicted in FIG. 1that those faces of adjacent cubes which are directly opposed or abuttedagainst one another have to be mirror images of one another before theedge color criteria is met. That is, all correspondingly opposed playingedges of those faces which are directly opposed to one another aregoverned by the same color. While this is true with respect to thesolution shown in FIG. 1, it is not necessarily true of other solutions.

FIG. 3 illustrates twelve other solutions meeting the criteria of theinvention which are possible with the preferred embodiment. Although notall faces of the cubes are depicted in the solutions depicted in FIG. 3,the faces that are depicted uniquely define the cube arrangements. Theedge relationship of all adjacent exposed faces which are not depictedin FIG. 3 also meet the solution criteria and, thus, each individualfigure in FIG. 3 represents a full disclosure of the solution with whichit is concerned.

As mentioned previously, it is only the edges which are defined byexposed faces which should meet the solution criteria other adjacentedges need not. For example, with reference to the solution depicted at11a, the four edges of the cubes which all meet at 36 and extenddownward through the puzzle are not all defined by the same color. Inthis connection, it will be seen that the faces on each cube whichdefine each of the edges meeting along the central line extendingthrough the puzzle at 36 are hidden within the pattern. There are onlythree edges on each of such faces which are also defined by faces whichare exposed, and therefore must meet the solution criteria.

It should be noted that for four cubes there are only thirteen differentgeometrical arrangements in which they can be placed joining one anotherwith regard to the number and position of hidden faces. Thus FIGS. 1 and3 represent solutions meeting the criteria of the invention for allthirteen of such arrangements.

The various solutions to the puzzle depicted in FIG. 1 and FIG. 3 differin the difficulty with which they are discoverable. In most cases, theeasiest solution to find is that depicted in FIG. 11 j in which none ofthe faces are hidden and the cubes join one another only at edges. Notethat the faces which are directed inwardly toward the center of thepattern of llj also meet the solution criteria since they are exposed.The most difficult solution to find is that depicted at 110. Either ahigh degree of skill or a considerable time is required to find thissolution which, incidently, is generally about as difficult or timeconsuming to find as the solution of the previously mentioned InstantInsanity puzzle. The solution of FIG. 1 as well as the other solutionsdepicted in FIG. 3 range in difficulty between the solutions 1 1 j and 11a. It will further be appreciated that the solution to FIG. 1 can beachieved easily when the mirror faces matching concept is discovered orpointed out, though its solution by the trial and error method can beextremely time consuming.

The provision of a puzzle of this nature having a plurality of differentsolutions assures that the puzzle will remain of interest to potentialplayers for a considerable period of time. Moreover, because thesolutions differ in the degree of skill or time required to find them,it will stimulate the interest of players of varying degrees of skill.

Although the invention has been described in connection with a preferredembodiment thereof, it will be appreciated by those skilled in the artthat various modifications can be made without departing from itsspirit. It is therefore intended that the coverage afforded be limitedonly by the terms of the claims and their equivalents.

I claim:

1. A puzzle comprising a plurality of geometrically similarthree-dimensional playing pieces, each one of which has a plurality ofintersecting faces providing playing edges for the piece, each of saidfaces having associated with each playing edge defined thereby one of aset of different indicia with the indicia associated with eachrespective edge of each of said faces being the same as the indicia onthe other of said faces associated with said respective edge, the orderof said indicia around the edges of each of said faces representing adifferent one of the mathematical combinations into which said indiciaare arrangeable with replacement in a circular order and the totalnumber of said different faces provided by said pieces equalling thetotal number of said different mathematical combinations into which saidindicia are arrangeable with replacement in such a circular order; andwherein a face of a first of said pieces is the mirror image of a firstface of a second of said pieces, a face of said second piece oppositesaid first face is the mirror image of a first face of a third of saidpieces, and a face of said third piece opposite said first face of saidthird piece is the mirror image of a face of a fourth of said pieces,whereby said playing pieces are arrangeable in a row with the indiciaassociated with all adjacent playing edges of different playing piecesmatching one another in a predetermined manner.

2. The puzzle of claim 1 wherein said pieces are arrangeable into ageometrical pattern in which faces of adjacent pieces are directlyopposed to one another and the indicia associated with allcorrespondingly opposed playing edges of said faces are matched in apredetermined manner.

3. The puzzle of claim 1 wherein said pieces provide twenty-fourdifferent faces defining said edges and wherein there are threedifferent indicia in said set of indicia, the order of said indiciaaround the edges of each of said faces representing a different one ofthe possible combinations into which said three indicia can be placed ina closed order with replacement when taken four at a time.

4. The puzzle of claim 3 wherein each of said playing pieces is a cubehaving six of said faces and there are four of said cubes providing saidtwenty-four faces.

5. The puzzle of claim 1 wherein each of said indicia is one of a set ofthree different colors.

1. A puzzle comprising a plurality of geometrically similarthree-dimensional playing pieces, each one of which has a plurality ofintersecting faces providing playing edges for the piece, each of saidfaces having associated with each playing edge defined thereby one of aset of different indicia with the indicia associated with eachrespective edge of each of said Faces being the same as the indicia onthe other of said faces associated with said respective edge, the orderof said indicia around the edges of each of said faces representing adifferent one of the mathematical combinations into which said indiciaare arrangeable with replacement in a circular order and the totalnumber of said different faces provided by said pieces equalling thetotal number of said different mathematical combinations into which saidindicia are arrangeable with replacement in such a circular order; andwherein a face of a first of said pieces is the mirror image of a firstface of a second of said pieces, a face of said second piece oppositesaid first face is the mirror image of a first face of a third of saidpieces, and a face of said third piece opposite said first face of saidthird piece is the mirror image of a face of a fourth of said pieces,whereby said playing pieces are arrangeable in a row with the indiciaassociated with all adjacent playing edges of different playing piecesmatching one another in a predetermined manner.
 2. The puzzle of claim 1wherein said pieces are arrangeable into a geometrical pattern in whichfaces of adjacent pieces are directly opposed to one another and theindicia associated with all correspondingly opposed playing edges ofsaid faces are matched in a predetermined manner.
 3. The puzzle of claim1 wherein said pieces provide twenty-four different faces defining saidedges and wherein there are three different indicia in said set ofindicia, the order of said indicia around the edges of each of saidfaces representing a different one of the possible combinations intowhich said three indicia can be placed in a closed order withreplacement when taken four at a time.
 4. The puzzle of claim 3 whereineach of said playing pieces is a cube having six of said faces and thereare four of said cubes providing said twenty-four faces.
 5. The puzzleof claim 1 wherein each of said indicia is one of a set of threedifferent colors.